Headteachers and local authorities say they are having to divert resources and funds from school budgets to pay for new kitchens as a result of the Liberal Democrats' underfunded policy of giving free school meals to all infants.

While Westminster was enthralled by the eruption over the policy between Lib Dem leader Nick Clegg and former Department for Education special adviser Dominic Cummings – with accusations flying that Clegg wanted Cummings charged under the official secrets act – school heads say they have been left to fend for themselves in parts of the country.

In Lincolnshire – a large, rural county with few commercial caterers to fall back on – the local authority has concluded that when the policy is introduced in September "it is clear that the county is likely to be under-resourced to meet need".

Lincolnshire's internal documents show a large shortfall in provision in parts of the county. The South Holland local government district in the south of the county was worst off with a forecast gap of 2,800 between pupil numbers and meal production, while West Lindsay district is short by almost 1,000.

"Lincolnshire is not in a good position," said the head of one school, who didn't want to be named. He is having to find more than £10,000 from existing budgets to prepare for the additional free meals for reception, year one and year two classes.

"The local authority said at a meeting with heads that they saw it as a series of races. One which is a sprint to get ready for September, and the marathon afterwards to make it sustainable. They want to spend the main amount of money from the government on the marathon. They didn't suggest how anyone was going to be able to make September work."

A teacher in rural Derbyshire said schools in his area had similar problems: "A school in London that has its own kitchens or providers close to it, those authorities seem to have got the same funding per capita as us.

"No one has looked at Derbyshire and said, it's a big rural authority, with schools with no kitchens, they are going to need £5m."

A spokesman for the DfE said: "Schools are receiving a huge amount of support to provide free school meals. We are providing £150m to improve kitchens and dining facilities, as well as an additional £22.5m specifically to help smaller schools to provide these meals."

In a joint article this week, education secretary Michael Gove and schools minister David Laws wrote: "At present the overwhelming majority seem to be on track and we are confident the policy will be delivered on time and on budget."

The ministers' confidence flies in the face of government calculations, uncovered last month, that said at least 2,700 school kitchens needed upgrades in order to be ready, and that 1,700 schools lacked a kitchen entirely.

In Bradford, the council is having difficulty finding the estimated £1.2m shortfall needed to upgrade equipment at 40 schools. The work may cost as much as £2.5m, but the Department for Education (DfE) is only providing £1.3m.

Ralph Berry, Bradford council's executive member for children's services, said the scheme lacked consultation between central and local government.

"It has been dropped into our laps to try and make it work. The objectives are laudable but the timescale and implementation is nothing short of chaotic," Berry said. He said the shortfall in capital funding provided by government was a "long way short" and Bradford may have to raid budgets designated for other projects to meet the extra costs.

The National Association of Head Teachers (NAHT) said that for the policy to succeed it was essential for schools to have kitchens and staff with the capacity to provide meals on a daily basis.

For many pupils their first experience of the policy will be cold lunches of sandwiches and similar prepared food, until kitchen improvements are completed, or through meals delivered by external caterers or at neighbouring schools.

In Worcestershire, the local authority said only 50 of 181 first, primary or special schools and academies in the county had fully operational kitchens able to cook hot meals. Of the 131 that did not have a full kitchen, the council said 126 would have some form of serving facilities so that meals could be sourced from outside caterers, leaving five still without any kitchen facilities at all.

The rushed implementation of the policy has also lead to fears that schools with a high proportion of pupils from low-income households could suffer, since their £1,300 pupil premium funding is tied to applications by parents for free school meals.

With children receiving lunches automatically, there is no incentive for parents to fill out forms for free school meals – and no alternative mechanism has been put in place.